Finite Group Databasebeta

← #10831719

Subgroups — #10831719

Order 768.

1112 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5579C211yesmaximal, characteristic, complemented
2C3×#2580C211yesmaximal, characteristic, complemented
3C2×#5007C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic
5#71635C211yesmaximal, characteristic, complemented
6#69657C212yesmaximal
7#69639C212yesmaximal, complemented
8#69625C212yesmaximal, complemented
9#69606C212yesmaximal, complemented
10#69556C211yesmaximal, characteristic, complemented
11#69555C211yesmaximal, characteristic, complemented
12#24339—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×#946—21no—
16C22×#213C2211yescharacteristic
17#5579—24no—
18#5579—24no—
19C2×#896—22no—
20C2×D3×C4⋊C4—21no—
21C2×C4×D12C2212yes—
22C2×C4×D12C2211yescharacteristic
23C2×#870C2212yes—
24C2×#869C2211yescharacteristic
25C3×#399—21no—
26C3×#393—21no—
27C3×#390C2212yes—
28C3×#389C2212yes—
29C3×#385—22no—
30C3×#382C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2212yes—
33#5014—24no—
34#5007—24no—
35C2×#809C2212yes—
36C2×#809C2211yescharacteristic
37C2×#808C2211yescharacteristic
38C2×#807C2212yes—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40C2×#803C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#554—31no—
47C2×#474—31no—
48C2×#442—31no—
49#2580D311yescharacteristic, complemented
50#2059—32no—
51#2025—32no—
52#1993—31no—
53#1985—31no—
54#1879—32no—
55#1862—32no—
56#1804—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C22×C4×D3—22no—
63C22×Dic6—21no—
64C3×C4×Q8—22no—
65C3×C4×Q8—22no—
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4D411yescharacteristic
70C2×C3×C4⋊C4D412yes—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C42C2312yes—
77C2×C3×C42C2311yescharacteristic
78#946—28no—
79C2×#213—22no—
80C2×#213—22no—
81C2×#213Q814yescomplemented
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—22no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3C2312yes—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C22×C3⋊C8C2311yescharacteristic
90C22×C3⋊C8C2312yes—
91C22×C3⋊C8C2311yescharacteristic
92#896—216no—
93D3×C4⋊C4—28no—
94C4×D12—24no—
95C4×D12—22no—
96C4×D12—24no—
97C4×D12D414yes—
98#870—24no—
99#870—24no—
100#869—22no—
101#869D414yes—
102C3×#95—21no—
103C3×#95—21no—
104C3×#95—22no—
105#832—24no—
106#832—22no—
107#809—22no—
108#809—24no—
109#808—22no—
110#807—24no—
111C4⋊(C3⋊C8)—22no—
112#803—22no—
113C22×D8—31no—
114#554—64no—
115#554—64no—
116C2×C4⋊Q8—31no—
117C2×#125—32no—
118C2×C22⋊Q8—61no—
119C2×C22⋊Q8—62no—
120C2×C4×Q8C2×D311yescharacteristic
121C2×C4×D4—31no—
122C2×C4×D4—32no—
123C2×C4×D4C2×D311yescharacteristic
124C22×C4⋊C4—61no—
125#474—64no—
126#442—64no—
127C2×#107—32no—
128C2×#106—31no—
129C2×C4⋊C8—31no—
130C2×#102—32no—
131C2×#102—31no—
132C2×#102—32no—
133C2×#102—31no—
134C2×#97—31no—
135#399—21no—
136#393—21no—
137#390C2×D312yes—
138#389C2×D312yes—
139#385—22no—
140#382C2×D311yescharacteristic
141#340—32no—
142#340—32no—
143#340—32no—
144#336—32no—
145#336—32no—
146C24×C3—21no—
147C23×D3—21no—
148C2×C3×Q8—22no—
149C2×C3×Q8—42no—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4C2×Q812yes—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—42no—
159C22×C3×C4—82no—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—21no—
166C22×Dic3—22no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12C2×D412yes—
171C2×D12C2×Q812yes—
172C2×D12#4112yes—
173C2×C4×D3—24no—
174C2×C4×D3—28no—
175C2×C4×D3—44no—
176C2×C4×D3—82no—
177C2×C4×D3—82no—
178C2×Dic6—24no—
179C2×Dic6—82no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—82no—
187C3×C22⋊C4—44no—
188C3×C22⋊C4—42no—
189C3×C22⋊C4—82no—
190C3×C42—22no—
191C3×C42—22no—
192C3×C42C2×D412yes—
193C3×C42C2×D412yes—
194#213—28no—
195(C2×D3)⋊C4—28no—
196(C2×D3)⋊C4—24no—
197C4⋊Dic3—24no—
198C4⋊Dic3—22no—
199C4⋊Dic3—21no—
200C4⋊Dic3C2×D414yes—
201C4⋊Dic3C2×D412yes—
202Dic3⋊C4—28no—
203Dic3⋊C4—28no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8C2×Q812yes—
213C22×Q8—61no—
214C22×Q8#6711yescharacteristic
215C22×D4—31no—
216C22×D4C22×D311yescharacteristic
217C23×C4—21no—
218C23×C4—61no—
219C23×C4—62no—
220C2×D8—34no—
221C2×D8—62no—
222C2×D8—62no—
223C22×C8—31no—
224C22×C8—31no—
225C22×C8—32no—
226C4⋊Q8—34no—
227C4⋊Q8—62no—
228#125—64no—
229#125—64no—
230C22⋊Q8—616no—
231C22⋊Q8—68no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—34no—
237C4×D4—64no—
238C4×D4—64no—
239C4×D4—62no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—32no—
243C2×C4⋊C4—68no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—61no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4#6711yescharacteristic
248C2×C4⋊C4#6712yes—
249C2×C4⋊C4C22×D311yescharacteristic
250C2×C4⋊C4C22×D312yes—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—61no—
256C2×C42C22×D312yes—
257C2×C42C22×D311yescharacteristic
258#107—64no—
259#106—62no—
260C4⋊C8—62no—
261#102—64no—
262#102—62no—
263#102—62no—
264#102—64no—
265#97—62no—
266#95—21no—
267#95—21no—
268#95—22no—
269C23×C3—22no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3C22×D411yescharacteristic
275C22×D3—22no—
276C22×D3—22no—
277C22×D3—22no—
278C22×D3—42no—
279C22×D3—42no—
280C3×Q8—44no—
281C3×D4—22no—
282C3×D4—22no—
283C3×D4—41no—
284C3×D4—41no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—84no—
299C2×C3×C4#12012yes—
300C2×C3×C4#12012yes—
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C22×Q811yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C42−1+411yescharacteristic
306C2×C3×C42−1+411yescharacteristic
307C2×Dic3—22no—
308C2×Dic3—24no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312D12—22no—
313D12—22no—
314D12—22no—
315D12C22⋊Q814yes—
316C4×D3—48no—
317C4×D3—88no—
318Dic6—84no—
319C3⋊C8—22no—
320C3⋊C8—41no—
321C3⋊C8—42no—
322C24—21no—
323C24—61no—
324C2×Q8—22no—
325C2×Q8—42no—
326C2×Q8—64no—
327C2×Q8—242no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—32no—
334C2×D4—32no—
335C2×D4—32no—
336C2×D4—61no—
337C2×D4—61no—
338C2×D4—61no—
339C2×D4D3×Q812yes—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—42no—
343C22×C4—68no—
344C22×C4—61no—
345C22×C4—62no—
346C22×C4—64no—
347C22×C4—82no—
348C22×C4—124no—
349C22×C4—242no—
350C22×C4—242no—
351C22×C4C2×#6712yes—
352C22×C4C2×#6712yes—
353C22×C4C2×#6711yescharacteristic
354C22×C4C2×#6711yescharacteristic
355C22×C4C23×D311yescharacteristic
356D8—68no—
357C2×C8—32no—
358C2×C8—61no—
359C2×C8—61no—
360C2×C8—61no—
361C2×C8—62no—
362C2×C8—62no—
363C2×C8—61no—
364C2×C8—62no—
365C2×C8—61no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—32no—
373C4⋊C4—34no—
374C4⋊C4—68no—
375C4⋊C4—68no—
376C4⋊C4—64no—
377C4⋊C4—61no—
378C4⋊C4—62no—
379C4⋊C4—82no—
380C22⋊C4—44no—
381C22⋊C4—42no—
382C22⋊C4—68no—
383C22⋊C4—64no—
384C22⋊C4—82no—
385C42—22no—
386C42—22no—
387C42D3×D412yes—
388C42D3×D412yes—
389C22×C3—44no—
390C22×C3—42no—
391C22×C3—82no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3C2×#12011yescharacteristic
395C22×C3#54811yescharacteristic
396C22×C3D4×Q811yescharacteristic
397C22×C3#57512yes—
398C22×C3#57612yes—
399C2×D3—24no—
400C2×D3—44no—
401C2×D3—44no—
402C2×D3—42no—
403C2×D3—42no—
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—82no—
409C3×C4—81no—
410C3×C4C2×C22⋊Q811yescharacteristic
411C3×C4#53611yescharacteristic
412C3×C4#54811yescharacteristic
413C3×C4D4×Q811yescharacteristic
414Dic3—42no—
415Dic3—44no—
416C23—22no—
417C23—41no—
418C23—42no—
419C23—41no—
420C23—42no—
421C23—62no—
422C23—62no—
423C23—62no—
424C23—122no—
425C23—122no—
426C23C22×#6711yescharacteristic
427Q8—44no—
428Q8—244no—
429D4—22no—
430D4—22no—
431D4—34no—
432D4—41no—
433D4—41no—
434D4—62no—
435D4—62no—
436D4—62no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—62no—
451C2×C4—64no—
452C2×C4—84no—
453C2×C4—128no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—248no—
458C2×C4#93712yes—
459C2×C4#93712yes—
460C2×C4C2×D3×D411yescharacteristic
461C2×C4C2×D3×D411yescharacteristic
462C2×C4C2×D3×Q811yescharacteristic
463C2×C4#100711yescharacteristic
464C2×C4D3×#4111yescharacteristic
465C2×C4#101011yescharacteristic
466C8—62no—
467C8—122no—
468C8—121no—
469C2×C3—82no—
470C2×C3#240911yescharacteristic
471C2×C3#300511yescharacteristic
472C2×C3#300811yescharacteristic
473C2×C3#337012yes—
474C2×C3#338912yes—
475D3—44no—
476C22—42no—
477C22—44no—
478C22—64no—
479C22—82no—
480C22—81no—
481C22—81no—
482C22—124no—
483C22—124no—
484C22—122no—
485C22—122no—
486C22C2×#93711yescharacteristic
487C22Q8×#6711yescharacteristic
488C22#582212yes—
489C22#583111yescharacteristic
490C22#583512yes—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—42no—
495C4—82no—
496C4—81no—
497C4—124no—
498C4—122no—
499C4D3×C22⋊Q811yescharacteristic
500C4#562811yescharacteristic
501C4#563211yescharacteristic
502C4#563311yescharacteristic
503C3#2433911yescharacteristic, complemented, Sylow
504C2—82no—
505C2—124no—
506C2#8275111yescharacteristic
507C2#8292811yescharacteristic
508C2#8338311yescharacteristic
509C2#8363312yes—
510C2#8371412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2580C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#2580D3
hypocenterC3#24339
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#5579C2semidirect
2. C3×#2580C2semidirect
3. C2×#5007C2semidirect
5. #71635C2semidirect
7. #69639C2semidirect
8. #69625C2semidirect
9. #69606C2semidirect
10. #69556C2semidirect
11. #69555C2semidirect
49. #2580D3semidirect
81. C2×#213Q8semidirect
503. C3#24339semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#55792
C3×#25802
C2×#50072
C2×#50142
#716352
#696572
#696392
#696252
#696062
#695562
#695552
#243393